Lenka Zdeborová sent me the following:
Surely you will find this paper yourself as you always do ... but let me send a link anyway: http://arxiv.org/abs/1301.5898 . We think that this is a nice contribution to the matrix factorization jungle ...! For calibration (and others, completion etc.) the algorithm works for a number of samples just a bit larger than the trivial counting bound, which is much much lower than anything else we have seen. It needs some tuning to work really well in the dictionary learning case, but we think this is a very promising track.At the same time, and again, if we missed some crucial references on the topic, please let us know.Best!
Thanks Lenka ! Here is the paper: Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning by Florent Krzakala, Marc Mézard, Lenka Zdeborová. The abstract reads:
We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.
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